Field-dependent collective ESR mode in YbRh2Si2

ARTICLE IN PRESS Physica B 404 (2009) 2964–2968

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Field-dependent collective ESR mode in YbRh2Si2 L.M. Holanda a, J.G.S. Duque a, E.M. Bittar a, C. Adriano a, P.G. Pagliuso a, C. Rettori a,, R.W. Hu b, C. Petrovic b, S. Maquilon c, Z. Fisk d, D.L. Huber e, S.B. Oseroff f a

Instituto de F´ısica ‘‘Gleb Wataghin’’, UNICAMP, C. P. 6165, 13083-970 Campinas, SP, Brazil Condensed Matter Physics, Brookhaven National Laboratory, Upton, NY 11973-5000, USA c Department of Physics, UC Davis, Davis, CA 95617, USA d University of California, Irvine, CA 92697-4574, USA e University of Wisconsin, Department of Physics, 1150 University Avenue, Madison, WI 53706, USA f San Diego State University, San Diego, CA 92182, USA b

a r t i c l e in f o

a b s t r a c t

PACS: 71.27.þa 75.20.Hr 76.30.v

Electron spin resonance (ESR) experiments in YbRh2Si2 Kondo lattice ðTK C25 KÞ at different field/ 3þ spin–lattice frequencies ð4:1rnr34:4 GHzÞ and H?c revealed: (i) a strong field dependent Yb relaxation, (ii) a weak field and T-dependent effective g-value, (iii) a suppression of the ESR intensity beyond 15% of Lu-doping, and (iv) a strong sample and Lu-doping ðr15%Þ dependence of the ESR data. 3þ These results suggest that the ESR signal in YbRh2Si2 may be due to a coupled Yb -conduction electron resonant collective mode with a subtle field-dependent spins dynamic. & 2009 Elsevier B.V. All rights reserved.

Keywords: Heavy fermions YbRh2Si2 Electron spin resonance (ESR)

1. Introduction The heavy-fermion (HF) Kondo lattice YbRh2Si2 ðTK C25 KÞ is an antiferrromagnetic (AF, TN ¼ 70 mK) tetragonal (I4/mmm) intermetallic compound. At low-T ðTtTK Þ the magnetic susceptibility exhibits a HF behavior and at high-T ðT\200 KÞ an anisotropic 3þ magnetic moment ðmeff C4:5mB Þ is Curie–Weiss with a full Yb observed [1–3]. The AF ordering of YbRh2Si2 may be driven to TN  0 by fields of H?c 650 Oe and HJc 7 kOe [4]. At these fields a quantum critical point (QCP) is observed with non-Fermi-liquid (NFL) behavior [1,2]. Therefore, among other systems [5,6], YbRh2Si2 is particularly an interesting system to study quantum criticality and NFL behavior. Electron spin resonance (ESR) experiments at low-T ðTt20 KÞ in YbRh2Si2 by Sichelschmidt et al. [7] have reported on a narrow ð1002200 OeÞ single dysonian resonance with no hyperfine components, T-dependence of the linewidth, DH, and a g-value 3þ anisotropy consistent with Yb in a metallic host of tetragonal symmetry. As in the early work of Tien et al. [8] the observation of 3þ ESR in the intermediate valence compound of a narrow Yb YbCuAl and, recently, in a dense Kondo system below TK were unexpected results [7]. Nevertheless, various reports were already 3þ in stoichiometric YbRh2Si2 [9], published on the ESR of Yb

 Corresponding author.

E-mail address: rettori@ifi.unicamp.br (C. Rettori). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.024

YbIr2Si2 [10], and YbRh2Si2 doped with non-magnetic impurities as Ge [11] and La [12]. Also, the ESR of Ce3þ in dense Kondo systems was communicated [13]. However, it is not clear yet what 3þ mechanism allows the observation of the Yb ESR line with local magnetic moment features in these highly correlated electron systems. The main purpose of this work is to investigate the H-dependent ESR data in the NFL phase of YbRh2Si2 (4:2tTt10 K; 0oHt10 kOe). We found an unexpected 3þ ESR data in YbRh2Si2 that, H-dependence behavior of the Yb we hope, will contribute to the understanding of the observed ESR signal in this system.

2. Experiment Single crystals of Yb1x Lux Rh2 Si2 (0rxt1:00) were grown from In and Zn-fluxes as reported elsewhere [14–16]. The structure and phase purity were checked by X-ray powder diffraction. The high quality of our undoped crystals was confirmed by X-rays rocking curves which revealed a mosaic structure of maximum c-axis angular spread of t0:0153 . The electrical residual resistivity ratio, r300 K =r1:9 K , for the In and Zn-flux grown crystals were 35 and 10, respectively [14–16]. For the ESR spectra 2  2  0:5 mm3 single crystals were used. The ESR experiments were carried out in a Bruker S, X, and Q-bands (4.1, 9.5 and 33:8 GHz) spectrometer using appropriated

ARTICLE IN PRESS L.M. Holanda et al. / Physica B 404 (2009) 2964–2968

3. Results and discussions Fig. 1 shows the YbRh2Si2 ESR X-band spectra at 4.2 K and H?c for single crystals grown in In and Zn-fluxes. From the anisotropy of the field for resonance, Hr ðyÞ (not shown), one can obtain the angular-dependence of the effective g-value which is given by 2 2 cos2 y þ gJc sin2 y1=2 . From the fitting of hn=mB Hr ðyÞ ¼ gðyÞ ¼ ½g?c the experimental Hr ðyÞ we obtain gJc t0:6ð4Þ and g?c ¼ 3:60ð7Þ. The inset of Fig. 1 displays, for H?c , the Korringa-like [18,19] linear thermal broadening of the linewidth, DHðTÞ ¼ a þ bT, and the fitting parameters for the crystal grown in Zn-flux at X and Q-bands. As it will be shown below, the large values measured for a and b indicates that Zn impurities were incorporated in this crystal. Fig. 2 presents the relative normalized X-band integrated ESR intensities for Yb1x Lux Rh2 Si2 at 4.2 K and H?c as a function of x, I4:2 ðxÞ=I4:2 ð0Þ. The ESR intensities were determined taking into consideration the crystal exposed area, skin depth and spectrometer conditions. These results show that, while for 3þ ESR intensity is nearly constant, for xr0:15 the Yb 0:15oxt1:00 the intensity vanish completely and no ESR could be detected. It is worth mention that for x40:15 and T\200 K, w?c ðTÞ follows an Curie–Weiss law with a full Yb3þ magnetic moment and that for xr0:15 there is no appreciable changes in the thermodynamic properties of these compounds [14,15]. The absence of resonance for x40:15 strongly suggests that the 3þ observed ESR for xo0:15 cannot be associated to a single Yb ion resonance but rather to a resonant collective mode of exchange 3þ coupled Yb –ce (conduction electrons) magnetic moments. We 3þ argue that a strong Yb –ce exchange coupling may broadens and

3þ

shifts the ce resonance toward the Yb resonance allowing their 3þ overlap and building up a Yb –ce coupled mode with possibly bottleneck/dynamic-like features. Evidences for bottleneck/ dynamic-like features in the Lu-doped crystals will be published 3þ local moments elsewhere. An internal field caused by the Yb may be responsible for the shift of the ce resonance [21]. Moreover, the Lu-doping may disrupt the collective mode coherence and, probably, may also opens the bottleneck/ dynamic regime [22]. Figs. 3a and b show, respectively, the low-T dependence of DH 3þ and effective g-value of the Yb ESR in In-flux grown YbRh2Si2,

1.6

I4.2 (x) / I4.2 (0)

resonators and T-controller systems. A single anisotropic resonance, with no hyperfine components, from the Kramer doublet ground state was observed at all bands. The dysonian lineshape ðA=B  2:5Þ corresponds to a microwave skin depth smaller than the size of the crystals [17].

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1.2

0.8

0.4 Yb1-xLuxRh2Si2

0.0 0

5

10

15

20

Lu x (%) 3þ

Fig. 2. X-band Yb relative normalized integrated ESR intensities, I4:2 ðxÞ=I4:2 ð0Þ, at 4.2 K and H?c for Yb1x Lux Rh2 Si2.

Abs. Der. (arb. units)

YbRh2Si2 T = 4.2 K 9.5 GHz

In-flux

In-flux (Ref. 7) Zn-flux

0

1

2 H (kOe)

3

4

Fig. 1. (Color online) ESR spectra at 4.2 K and H?c for single crystals grown in In and Zn-fluxes and for the In-flux crystals of Ref. [7]. Inset: DHðTÞ ¼ a þ bT for Znflux crystals and H?c at X and Q-bands.

3þ

Fig. 3. (Color online) X, S and Q-bands low-T dependence of the Yb ESR for H?c : (a) DHðTÞ and (b) effective gðTÞ-value. The lines represent the linear fit in (a) and a guide to the eyes in (b).

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measured at S, X and Q-bands for H?c. In this T-interval, and within the error bars, it is also found that DH ¼ a þ bT for the three bands. This suggests a Korringa-type of mechanism for the 3þ 3þ Yb spin–lattice relaxation (SLR), i.e. the Yb local moment is exchange coupled to the conduction electrons [18]. The residual linewidth, a, and relaxation-rate, b ¼ DH=DT, are given in Fig. 3a. Notice that the minimum relaxation rate, b, is found at X-band, HC1900 Oe. The actual determination of the residual linewidth, a ¼ DHðT ¼ 0Þ, would require measurements at lower-T, therefore, the obtained values should be consider just as fitting parameters. A H-dependent SLR-rate b is not expected for a normal local magnetic moment-ce exchange coupled system, where the Korringa-rate is frequency/field independent [19]. However, 3þ since the Yb and ce magnetic moments carry unlike spins and different g-values, the H-dependence of b may be an anomalous manifestation of a bottleneck-like behavior. Fig. 3b shows that the T-dependence of the effective g-values are slightly different in the three bands, with minimum effective g4:2 -values also at the X-band. The effective g-value accuracy is much higher than that 2 2 cos2 y þ gJc sin2 y1=2 obtained from hn=mB Hr ðyÞ ¼ gðyÞ ¼ ½g?c because proper experimental conditions were chosen for these H?c measurements. Fig. 4 displays the H-dependence of b and g4:2 -values. Notice that both parameters have minimum values at the X-band field, HC1900 Oe.

Fig. 4. (Color online) Yb g4:2 -value for H?c .

3þ

Fig. 5 shows the X-band DHðTÞ for 4:2rTr21 K and H?c . The data were fitted to DHðTÞ ¼ a þ bT þ cd=½expðd=TÞ  1 taking into consideration all the contributions to DH in a metallic host. The 1st and 2nd terms are the same as above. The 3rd is the relaxation, also via an exchange interaction with the ce, of a thermally 3þ populated Yb excited crystal field state at d K above the ground state [23]. The fitting parameters are in the inset of Fig. 5. This 3þ spin–phonon analysis does not consider any direct Yb contribution [23]. The S and Q-band DHðTÞ data for 7tTt20 K also show exponential behaviors with c  200ð70Þ Oe=K and d  75ð20Þ K. Within a molecular field approximation the effective gðTÞ may be written as geff ¼ gð1 þ lw?c ðTÞÞ. Fig. 6 presents a plot of Dg=g ¼ ðgeff ðTÞ  gð15ÞÞ=gð15Þplw?c ðTÞ for Tr15 K and X-band for our x ¼ 0 crystals [15] and that from Refs. [7,24]. A linear correlation is obtained with l values in the interval of 2 kOe=mB 4l4  3 kOe=mB . In the Appendix, it is shown that the shift that gives the temperature dependence of geff arises from 3þ anisotropic exchange interactions between the Yb ions. Fig. 7 shows the comparison between theory and experiment. Therefore, these results definitely indicate that the T-dependence of the effective gðx; TÞ is nothing but a consequence of the shift of the

ESR H-dependence (X, S and Q-bands) of b and effective

Fig. 6. (Color online) effective g-shift and magnetic susceptibility correlation, ðgeff  gÞ=gplw?c ðTÞ, for Tr15 K and X-band for x ¼ 0 crystals (see text).

Fig. 5. (Color online) Yb ESR X-band DHðTÞ for H?c and 4:2rTr21 K. The fitting parameters correspond to the best fit to DHðTÞ ¼ a þ bT þ cd=½expðd=TÞ  1.

Fig. 7. g?c ðTÞ vs. T. The solid line is the theoretical curve obtained from Eq. (4). The inset shows the fit of w1 ?c ðTÞ to a Curie–Weiss law.

3þ

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field for resonance toward higher fields due to an AF internal molecular field and has nothing to do with a Dg ¼ c=lnðTK =TÞ divergence [7]. Moreover, the expected g-shift caused by the 3þ exchange interaction between the Yb and ce local moments, Jfce , can be estimated from the largest Korringa-rate value measured at Q-band in our In-flux crystal, b ffi 45 Oe=K. Within a single band 3þ approximation [23] and absence of q-dependence of the Yb –ce exchange interaction, Jfce ðqÞ ffi Jfce ð0Þ, [26] one can write ðDg=gÞ2 ¼ mB b=pgkB , which givesjDg=gjt2%. This values is far much smaller than that estimated in Ref. [7] using as a 3þ reference the ESR of Yb in the insulator PbMoO4. On the other hand, from the Korringa relation [23] using b ffi 45 Oe=K and assuming a maximum bare density of state per one spin direction at the Fermi level, ZF , given by the Somerfield coefficient of the specific heat measurements ðg ffi 900 mJ=mol K2 Þ [27] we extract a lower limit for jJfce j\3 meV, which is about two order of 3þ 3þ magnitude larger than the value found for the Yb –Yb exchange interaction, jJff j, inferred from the molecular field parameters yJc and y?c ðTÞ in a nearest-neighbor approximation (see Appendix) [25]. Our experiments confirm the ESR results of Sichelschmidt et al., in YbRh2Si2 below TK C25 K [7]. However, our results suggest that this ESR corresponds to a strong exchange coupled 3þ Yb –ce resonant collective mode. The features of this resonant collective mode resembles the bottleneck/dynamic scenario for diluted magnetic moments exchange coupled to the ce, both with g ffi 2, where in a normal metal the SLR-rate, b, and g-value depend on the competition between the Korringa/Overhauser relaxation and the ce SLR [18,29,19,28]. Then, the increase of b by the addition of non-magnetic impurities to YbRh2Si2 (Lu, Zn in Fig. 1 and La in Ref. [12]), may be associated to ‘‘opening’’ the bottleneck regime due to the increase in the ce spin-flip scattering [19,28]. The results and discussion about the non-magnetic Lu impurities effects and bottleneck behavior will be the subject of a forthcoming publication. Another striking result reported in Fig. 4 is the non-monotonic 3þ H-dependence of b and effective g-value of the Yb –ce resonant collective mode. Admixtures via Van Vleck terms [30] may be disregarded because this contribution should scale with H. Therefore, we believe that the low H-tunability [1–3] of the ESR parameters in YbRh2Si2 is an ‘‘intrinsic’’ property of the NFL state 3þ near a QCP, where the strength of the Yb –ce magnetic coupling may be subtly tuned and allows the formation of the resonant collective mode. We attribute the absence of low H-dependent ESR results in previous reports [7] to the presence of ‘‘extrinsic’’ impurities and/or Rh/Si defects [15,31] that increase the SLR, b, and residual linewidth, a. Thus, as for our Zn-flux crystals, hiding the low H-dependence of the ESR parameters in the NFL phase. 3þ In the bottleneck scenario, the Yb –ce resonant collective mode presents the strongest bottleneck regime (smallest b) at H  1900 Oe. However, due to the subtle details of the coupling between the Kondo ions and the ce in a Kondo lattice and to strong impurity effects, these resonant collective modes may not be always observable, unless extreme bottleneck regime is achieved. The proximity to a QCP and/or the presence of enhanced spin susceptibility may favor this condition [13,32]. The bottleneck 3þ scenario for the Yb –ce resonant collective mode may also explain 3þ the absence of Yb hyperfine ESR structure [33]. Recent calculations by Abrahams and Wolfle have suggested that the ESR linewidth may be strongly reduced by a factor involving the heavy fermion mass and quasiparticle ferromagnetic ð0ÞÞ [34]. These (FM) exchange interactions ðm=m Þð1  U~ ~wþ ff ;H results indicate that the estimation of the linewidth from the Kondo temperature, TK ðDH ¼ kB TK =g mB Þ is an over estimation. However, these calculations may not be contemplating all the

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possibilities and have to be taken with care when applied to the dynamic of the ESR of YbRh2Si2 compound because, (i) it presents 3þ 3þ an AF Yb 2Yb exchange interaction (although other works in literatures have claimed in favor of the existence of FM fluctuation in YbRh2Si2 [35,36] and (ii) samples with the same thermodynamic properties present quit different linewidths (see Fig. 1). Furthermore, the anisotropy in the ESR in YbRh2Si2 reflects both 3þ 3þ 3þ and Yb –ce single-ion crystal field effects and the Yb –Yb interactions. In principle, the analysis of crystal field effects is straightforward although somewhat hindered by the inability to detect a signal when the field is along the c-axis. The anisotropy of 3þ 3þ 3þ the Yb –Yb and Yb –ce is more difficult to determine and in the latter case more critical. The application of the resonant 3þ collective mode model is based on the assumption that the Yb –ce 3þ coupling is dominated by a scalar interaction between the Yb ground state doublet pseudo-spins Sps , and the spins of the conduction electrons, s, with the consequence that the total spin s þ Sps is (approximately) a constant of the motion [37,38]. In the presence of uniaxial anisotropy, only the component of the total spin along the symmetry axis is a constant of the motion. How the lower symmetry affects the formation of the collective mode is an unsolved problem requiring further study [34]. Finally, we hope that our results motivate new theoretical approaches to understand the dynamics of strong exchange coupled magnetic moments of unlike spins and g-values, as 3þ and ce, and explore the general existence of a resonant Yb collective mode with a bottleneck/dynamic-like behavior.

4. Summary In summary, this work reports low H-dependent ESR, below TK C25 K, in the NFL phase of YbRh2Si2 ðTt10 KÞ. It is suggested 3þ that the observed ESR in YbRh2Si2 corresponds to a Yb –ce resonant collective mode in a strong bottleneck-like regime, which is highly affected by the presence of impurities and/or defects. The analysis of our data allowed us to give estimations for the 3þ 3þ exchange parameter, Jff , and a lower limit for the Yb 2Yb 3þ Yb –ce exchange parameter, jJfce j. Acknowledgments We thank FAPESP and CNPq (Brazil) and NSF (USA) for financial support; and P. Coleman, E. Miranda, D.J. Garcia and M. Continentino for fruitful discussions.

Appendix In Fig. 6, it was noted that geff depends linearly on the transverse susceptibility. In this appendix we show the origin of the linear dependence. The starting point is a general theory for magnetic resonance in anisotropic magnets [39] which leads to an expression for the effective g-factor in an orthorhombic crystal that can be written in the form: geff ¼

gbM gcM a M ga ð a b Þ1=2

w

ww

when the static field is along the a-axis. M g-factors are denoted by ga;b;c and the susceptibilities by wa;b;c . In the case of a the static field perpendicular to the c-axis, geff ?c ¼ gcM



w?c wJc

ð1Þ Here the microscopic corresponding static uniaxial system with (1) reduces to

1=2 ð2Þ

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In applying this equation to the resonance in YbRh2Si2 it must be kept in mind that the g-factors and susceptibilities refer to the ground state doublet. We assume that the relevant doublet susceptibilities take the form C?;J =ðT þ y?;J Þ where C? and CJ are constants and fit the experimental data for w?c to the form w0 þ C?c =ðT þ y?c Þ in the temperature range of the resonance experiment (see inset of Fig. 7) with the result y?c ¼ 1:48 K. We chose yJc 0 and the overall amplitude g?c as adjustable parameters. The measured values g?c ðTÞ were then fit to the function:    y?c  yJc 1=2 0 g?c ðTÞ ¼ g?c 1 ð3Þ T þ y?c    y?c  yJc 0 1  0:5  g?c T þ y?c

ð4Þ

0 with the best fit given by g?c ¼ 3:66 and yJc ¼ 1:09 K (see Fig. 7). The result shown in Fig. 6 is in accord with Eq. (4) and Fig. 7 since the T-dependent part of w?c ðTÞ is proportional to ðT þ y?c Þ1 , the same factor that is present in Eq. (4). The results outlined in the preceding paragraph show that the T-dependence of g?c ðTÞ is associated with the difference in the longitudinal and transverse exchange interaction which is reflected in the difference between y?c and yJc . Although it has not been possible to observe the resonance with the static field along the c-axis, there is a corresponding shift there as well, which takes the form:    y?c  yJc 0 ð5Þ 1þ gJc ðTÞ ¼ gJc T þ yJc

Note that the shift in gJc ðTÞ is in the opposite direction from the shift in g?c ðTÞ. References [1] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F.M. Grosche, P. Gegenwart, M. Lang, G. Sparn, F. Steglich, Phys. Rev. Lett. 85 (2000) 626. [2] P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama, K. Tenya, O. Trovarelli, F. Steglich, Phys. Rev. Lett. 89 (2002) 056402. [3] J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y. Tokiwa, O. Trovarelli, C. Geibel, F. Steglich, C. Pepin, P. Coleman, Nature (London) 424 (2003) 524. [4] S. Paschen, T. Luhmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman, Q. Si, Nature 432 (2004) 881. [5] R.H. Heffner, M.R. Norman, Comments Condens. Matter Phys. 17 (1996) 361. [6] F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, H. Schafer, Phys. Rev. Lett. 43 (1979) 1892.

[7] J. Sichelschmidt, V.A. Ivanshin, J. Ferstl, C. Geibel, F. Steglich, Phys. Rev. Lett. 91 (2003) 156401. [8] C. Tien, J.-T. Yu, H.-M. Duh, Jpn. J. Appl. Phys. 32 (1993) 2658. [9] J. Sichelschmidt, J. Wykhoff, H.-A. Krug von Nidda, J. Ferstl, C. Geibel, F. Steglich, J. Phys. Condens. Mater. 19 (2007) 116204. [10] J. Sichelschmidt, J. Wykhoff, H.-A. Krug von Nidda, I.I. Fazlishanov, Z. Hossain, C. Krellner, C. Geibel, F. Steglich, J. Phys. Condens. Mater. 19 (2007) 016211. [11] J. Sichelschmidt, J. Ferstl, C. Geibel, F. Steglich, Physica B 359–361 (2005) 17. [12] J. Wykhoff, J. Sichelschmidt, J. Ferstl, C. Krellner, C. Geibel, F. Steglich, I. Fazlishanov, H.-A. Krug von Nidda, Physica C 460–462 (2007) 686. ¨ [13] C. Krellner, T. Forster, H. Jeevan, C. Geibel, J. Sichelschmidt, Phys. Rev. Lett. 100 (2008) 066401. ¨ [14] U. Kohler, N. Oeschler, F. Steglich, S. Maquilon, Z. Fisk, Phys. Rev. B 77 (2008) 104412. [15] S. Maquilon, Ph.D thesis at UC Davis, 2007; Z. Fisk, Private communication. [16] R.W. Hu, J. Hudis, C. Stock, C.L. Broholm, C. Petrovic, J. Crystal Growth 304 (2007) 114. [17] G. Feher, A.F. Kip, Phys. Rev. 98 (1955) 337; F.J. Dyson, Phys. Rev. 98 (1955) 349; G.E. Pake, E.M. Purcell, Phys. Rev. 74 (1948) 1184. [18] J. Korringa, Physica 16 (1950) 601; H. Hasegawa, Progr. Theor. Phys. (Kyoto) 21 (1959) 1093. [19] C. Rettori, D. Davidov, H.M. Kim, Phys. Rev. B 8 (1973) 5335; C. Rettori, D. Davidov, R. Orbach, E.P. Chock, B. Ricks, Phys. Rev. B 7 (1973). [21] S. Oseroff, M. Passeggi, D. Wohlleben, S. Schultz, Phys. Rev. B (1977) 1283. [22] C. Rettori, H.M. Kim, E.P. Chock, D. Davidov, Phys. Rev. B 10 (1974) 1826. [23] G.E. Barberis, D. Davidov, J.P. Donoso, C. Rettori, J.F. Suassuna, H.D. Dokter, Phys. Rev. B 19 (1979) 5495. [24] J. Custers, P. Gegenwart, C. Geibel, F. Steglich, T. Tayama, O. Trovarelli, N. Harrison, Acta Phys. Pol. B 32 (2001) 3211. [25] P.G. Pagliuso, D.J. Garcia, E. Miranda, E. Granado, R. Lora-Serrano, C. Giles, J.G.S. Duque, R.R. Urbano, C. Rettori, J.D. Thompson, M.F. Hund, J.L. Sarrao, J. Appl. Phys. 99 (2006) 08P703. [26] D. Davidov, K. Maki, R. Orbach, C. Rettori, E.P. Chock, Solid State Commun. 12 (1973) 621. [27] P. Gegenwart, Y. Tokiwa, T. Westerkamp, F. Weickert, J. Custers, J. Ferstl, ¨ C. Krellner, C. Geibel, P. Kerschl, K.-H. Muller, F. Steglich, New J. Phys. 8 (2006) 171. [28] C. Rettori, D. Davidov, G. Ng, E.P. Chock, Phys. Rev. B 12 (1975) 1298. [29] A.W. Overhauser, Phys. Rev. 89 (1953) 689. [30] A. Abragam, B. Bleaney, EPR of Transition Ions, Clarendon Press, Oxford, 1970. [31] D. Louca, J.D. Thompson, J.M. Lawrence, R. Movshovich, C. Petrovic, J.L. Sarrao, G.H. Kwei, Phys. Rev. B 61 (2000) R14940. [32] P. Monod, J. Phys. 39 (1978) C6-1472. [33] D. Davidov, C. Rettori, R. Orbach, A. Dixon, E.P. Chock, Phys. Rev. B 11 (1975) 3546. [34] E. Abrahams, P. Wolfle, arXiv:0808.0892, unpublished. [35] K. Ishida, K. Okamoto, Y. Kawasaki, Y. Kitaoka, O. Trovarelli, C. Geibel, F. Steglich, Phys. Rev. Lett. 89 (2002) 107202. [36] P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett. 94 (2005) 076402. [37] D.L. Huber, Phys. Rev. B 12 (1975) 31; D.L. Huber, Phys. Rev. B 13 (1976) 291. [38] S.E. Barnes, J. Zitcova-Wilcox, Phys. Rev. B 7 (1973) 2163. [39] D.L. Huber, M.S. Seehra, Phys. Status Solidi (b) 74 (1976) 145.